4
Election schemes suppose there is an election between two parties, called 0 and 1 assume unrealistically that n voters cast votes independently and unif. randomly an election scheme is a boolean function f : {0,1} n → {0,1} mapping votes to winner what if there are errors in recording of votes? suppose each vote is misrecorded independently with prob. ε.

5

6
Election schemes suppose there is an election between two parties, called 0 and 1 assume unrealistically that n voters cast votes independently and unif. randomly an election scheme is a boolean function f : {0,1} n → {0,1} mapping votes to winner what if there are errors in recording of votes? suppose each vote is misrecorded independently with prob. ε. what is the prob. this affects elec.’s outcome?

16
History of Noise Sensitivity Kahn-Kalai-Linial ’88 The Influence of Variables on Boolean Functions

17
Kahn-Kalai-Linial ’88 implicitly studied noise sensitivity motivation: study of random walks on the hypercube where the initial distribution is uniform over a subset the question, “What is the prob. that a random walk of length εn, starting uniformly in f -1 (1), ends up outside f -1 (1)?” is essentially asking about NS ε (f) famous for using Fourier analysis and “Bonami-Beckner inequality” in TCS

19
Håstad ’97 breakthrough hardness of approximation results decoding the Long Code: given access to the truth-table of a function, want to test that it is “significantly” determined by a “junta” (very small number of variables) roughly, does a noise sensitivity test: picks x and y as in n.s., tests f(x)=f(y)

24
Hardness on average def: We say f : {0,1} n → {0,1} is (1-ε)-hard for circuits of size s if there is no circuit of size s which computes f correctly on more than (1-ε)2 n inputs. def: A complexity class is (1-ε)-hard for polynomial circuits if there is a function family (f n ) in the class such that for suff. large n, f n is (1-ε)-hard for circuits of size poly(n).

25
Hardness of EXP, NP Of course we can’t show NP is even (1-2 -n )- hard for poly ckts, since this is NP µ P/poly. But let’s assume EXP, NP µ P/poly. Then just how hard are these for poly circuits? For EXP, extremely strong results known – [ BFNW 93,Imp95,IW97,KvM99,STV99]: if EXP is (1-2 -n )-hard for poly circuits, then it is (½ + 1/poly(n))-hard for poly circuits. What about NP?

26
Yao’s XOR Lemma Some of the hardness amplification results for EXP use Yao’s XOR Lemma: Thm: If f is (1-ε)-hard for poly circuits, then PARITY k ­ f is (½+½(1-2ε) k )-hard for poly circuits. Here, if f is a boolean fcn on n inputs and g is a boolean fcn on k inputs, g ­ f is the function on kn inputs given by g(f(x 1 ), …, f(x k )). No coincidence that the hardness bound for PARITY k ­ f is 1-NS ε ( PARITY k ).

27
A general direct product thm. Yao doesn’t help for NP – if you have a hard function f n in NP, PARITY k ­ f n probably isn’t in NP. We generalize Yao and determine the hardness of g ­ f n for any g – in terms of the noise sensitivity of g: Thm: If f (balanced) is (1-ε)-hard for poly circuits, then g ­ f n is roughly (1-NS ε (g))- hard for poly circuits.

29
Hardness of NP If (f n ) is a (hard) function family in NP, and (g k ) is a monotone function family, then (g k ­ f n ) is in NP. We give constructions and prove tight bounds for the problem of finding monotone g such that NS ε (g) is very large (close to ½) for ε very small. Thm: If NP is (1-1/poly(n))-hard for poly ckts, then NP is (½ + 1/√n)-hard for poly ckts.

30
Learning algorithms

31
Learning theory Learning theory ([Valiant84]) deals with the following scenario: someone holds an n-bit boolean function f you know f belongs to some class of fcns (eg, {parities of subsets}, {poly size DNF}) you are given a bunch of uniformly random labeled examples, (x, f(x)) you must efficiently come up with a hypothesis function h that predicts f well

32
Learning noise-stable functions We introduce a new idea for showing function classes are learnable: Noise-stable classes are efficiently learnable Thm: Suppose C is a class of boolean fcns on n bits, and for all f ∈ C, NS ε (f) ≤ β(ε). Then there is an alg. for learning C to within accuracy ε in time: n O(1)/β (ε).

33
Example – halfspaces E.g., using [Peres98], every boolean function f which is the “intersection of two halfspaces” has NS ε (f) ≤ O(√ε). Cor: The class of “intersections of two halfspaces” can be learned in time n O(1/ε²). No previously known subexponential alg. We also analyze the noise sensitivity of some more complicated classes based on halfspaces and get learning algs. for them.

34
Why noise stability? Suppose a function is fairly noise stable. In some sense this means if you know f(x), you have a good guess for f(y) for y’s which are somewhat close to x in Hamming distance. Idea: Draw a “net” of examples: (x 1, f(x 1 )), … (x M, f(x M )). To hypothesize about y, compute a weighted average of known labels, based on dist. to y: hypothesis =… sgn[ w(Δ(y,x 1 ))f(x 1 ) + ··· + w(Δ(y,x M ))f(x M ) ].

36
Learning juntas The essential blocking issue for learning poly size DNF formulas is that they can be O(log n)-juntas. Previously, no known algorithm for learning k-juntas in time better than the trivial n k. We give the first improvement: algorithm runs in time n.704k. Can the strong relationship between juntas and noise sensitivity improve this?

41
“Cosmic coin flipping” n random votes cast in an election we use a balanced election scheme, f k different auditors get copies of the votes; however, each gets an ε-noisy copy what is the probability all k auditors agree on the winner of the election? Equivalently, k distributed parties want to flip a shared random coin given noisy access to a “cosmic” random string.

42
Relevance of the problem Application of this scenario: “Everlasting security” of [DingRabin01] – a cryptographic protocol assuming that many distributed parties have access to a satellite broadcasting stream of random bits. Also a natural error-correction problem: without encoding, can parties attain some shared entropy?

43
Success as function of k Most interesting asymptotic case: ε a small constant, n unbounded, k → ∞. What is the maximum success probability? Surprisingly, goes to 0 only polynomially: Thm: The best success probability of k players is Õ( 1/k 4ε ), with the majority function being essentially optimal.

44
Reverse Bonami-Beckner To prove that no protocol can do better than k -Ω(1), we need to use a reverse Bonami- Beckner inequality [Bor82]: for f ≥ 0, t ≥ 0, ||T λ (f)|| 1-t/λ ≥ ||f|| 1-tλ Concentration of measure interpretation: Let A be a reasonably large subset of the cube. Then almost all x have Pr[y ∈ A] somewhat large.